\displaystyle [\partial_r (f\circ \Phi), \partial_\varphi (f\circ \Phi), \partial_\psi (f\circ \Phi)]=\\
=[\partial_x f, \partial_y f, \partial_z f] \cdot\left[\begin{array}{ccc}
\cos\psi\cos\varphi & -r\cos\psi\sin\varphi & -r\sin\psi\cos\varphi\\
\cos\psi\sin\varphi & r\cos\psi\cos\varphi & -r\sin\psi\sin\varphi\\
\sin\psi & 0 & r\cos\psi
\end{array}\right]